Joint Higher-Rank Numerical Ranges

• Joint higher-rank numerical ranges are operator-theoretic sets defined by summing expectation values over orthonormal k-frames, captured through rank-k compressions.
• They admit several equivalent formulations—including projection and isometric forms—and link geometric properties to algebraic features like reducing subspaces and spectral data.
• The topic also explores convexity, closure, and polyhedrality, with applications extending to quantum error correction via joint matricial compressions and operator diagonalizations.

Joint higher-rank numerical ranges are operator-theoretic sets that encode summed expectation values of an $m$-tuple of bounded operators over orthonormal $k$-frames. For a complex Hilbert space ${\mathcal H}$, a positive integer $1 \le k < \dim {\mathcal H}$, and $${\mathbf A}=(A_1,\dots,A_m)\in {\mathcal B}({\mathcal H})^m$$, the joint $k$-numerical range is

$$W_k({\mathbf A})=\left\{(\alpha_1,\dots,\alpha_m)\in {\mathbb C}^m:\ \alpha_i=\sum_{j=1}^k \langle A_i x_j,x_j\rangle \text{ for an orthonormal set } \{x_1,\dots,x_k\}\subset {\mathcal H}\right\}.$$

Its study links the geometry of $W_k({\mathbf A})$ to algebraic features such as reducing subspaces, diagonal compressions, commutativity, normality, and essential spectral data; in infinite dimensions, closure and polyhedrality are controlled in part by the joint essential numerical range (Chan et al., 2021).

1. Definitions and equivalent formulations

The basic definition of $W_k({\mathbf A})$ admits several equivalent formulations. In projection form,

$$W_k({\mathbf A})=\{(\operatorname{Tr}(PA_1),\dots,\operatorname{Tr}(PA_m)):\ P=P^*=P^2,\ \operatorname{rank}P=k\}.$$

In isometric form,

$k$0

There is also a compression characterization: $k$1 if and only if there exists a unitary $k$2 and a $k$3 matrix tuple $k$4 with $k$5 such that

$k$6

These formulations make explicit that $k$7 is determined by rank-$k$8 compressions rather than by scalar restrictions (Chan et al., 2021).

In finite dimension $k$9, the endpoint cases are fixed by convention or trace identities: ${\mathcal H}$0 and

${\mathcal H}$1

The range is translation-covariant and linear-covariant: for ${\mathcal H}$2,

${\mathcal H}$3

and under linear recombination by a matrix ${\mathcal H}$4, one has

${\mathcal H}$5

If ${\mathcal H}$6 and each ${\mathcal H}$7 decomposes accordingly, then ${\mathcal H}$8 is constrained by convex combinations of sums of lower-rank joint numerical ranges on the summands; equality holds when ${\mathcal H}$9 is convex (Chan et al., 2021).

A persistent source of terminological ambiguity is the distinction between $1 \le k < \dim {\mathcal H}$0 and the higher-rank numerical range $1 \le k < \dim {\mathcal H}$1. For a single operator $1 \le k < \dim {\mathcal H}$2,

$1 \le k < \dim {\mathcal H}$3

and for tuples the analogous notion requires simultaneous scalar compressions. By contrast, $1 \le k < \dim {\mathcal H}$4 records summed expectations over orthonormal $1 \le k < \dim {\mathcal H}$5-tuples and generally differs from $1 \le k < \dim {\mathcal H}$6 unless strong structural conditions are present. In the matricial-range literature, the joint higher-rank numerical range $1 \le k < \dim {\mathcal H}$7 appears as the scalar slice of the joint higher-rank matricial range $1 \le k < \dim {\mathcal H}$8, and the more general joint $1 \le k < \dim {\mathcal H}$9-matricial range $${\mathbf A}=(A_1,\dots,A_m)\in {\mathcal B}({\mathcal H})^m$$0 compresses each $${\mathbf A}=(A_1,\dots,A_m)\in {\mathcal B}({\mathcal H})^m$$1 to $${\mathbf A}=(A_1,\dots,A_m)\in {\mathcal B}({\mathcal H})^m$$2 with diagonal $${\mathbf A}=(A_1,\dots,A_m)\in {\mathcal B}({\mathcal H})^m$$3 (Kribs et al., 2019).

2. Geometry and convexity

A central fact is that the joint setting departs sharply from the classical Toeplitz–Hausdorff picture. For a single operator, higher-rank numerical ranges are convex, but for $${\mathbf A}=(A_1,\dots,A_m)\in {\mathcal B}({\mathcal H})^m$$4 operators the joint $${\mathbf A}=(A_1,\dots,A_m)\in {\mathcal B}({\mathcal H})^m$$5-numerical range need not be convex. Chan, Li, and Poon formulate convexity criteria in terms of the affine span of $${\mathbf A}=(A_1,\dots,A_m)\in {\mathcal B}({\mathcal H})^m$$6. If $${\mathbf A}=(A_1,\dots,A_m)\in {\mathcal B}({\mathcal H})^m$$7, then $${\mathbf A}=(A_1,\dots,A_m)\in {\mathcal B}({\mathcal H})^m$$8 is convex if and only if $${\mathbf A}=(A_1,\dots,A_m)\in {\mathcal B}({\mathcal H})^m$$9 has dimension at most $k$0. If $k$1 and $k$2 has dimension at most $k$3, then $k$4 is convex; in particular, $k$5 is always convex. Conversely, if $k$6 and the span has dimension at least $k$7, there exists $k$8 such that $k$9 is not convex (Chan et al., 2021).

The Pauli matrices supply the canonical low-dimensional examples. With $$W_k({\mathbf A})=\left\{(\alpha_1,\dots,\alpha_m)\in {\mathbb C}^m:\ \alpha_i=\sum_{j=1}^k \langle A_i x_j,x_j\rangle \text{ for an orthonormal set } \{x_1,\dots,x_k\}\subset {\mathcal H}\right\}.$$0, one has $$W_k({\mathbf A})=\left\{(\alpha_1,\dots,\alpha_m)\in {\mathbb C}^m:\ \alpha_i=\sum_{j=1}^k \langle A_i x_j,x_j\rangle \text{ for an orthonormal set } \{x_1,\dots,x_k\}\subset {\mathcal H}\right\}.$$1 equal to the unit disk, while $$W_k({\mathbf A})=\left\{(\alpha_1,\dots,\alpha_m)\in {\mathbb C}^m:\ \alpha_i=\sum_{j=1}^k \langle A_i x_j,x_j\rangle \text{ for an orthonormal set } \{x_1,\dots,x_k\}\subset {\mathcal H}\right\}.$$2 is the unit sphere in $$W_k({\mathbf A})=\left\{(\alpha_1,\dots,\alpha_m)\in {\mathbb C}^m:\ \alpha_i=\sum_{j=1}^k \langle A_i x_j,x_j\rangle \text{ for an orthonormal set } \{x_1,\dots,x_k\}\subset {\mathcal H}\right\}.$$3, hence nonconvex. This contrast encapsulates the fact that adding one more coordinate can destroy convexity even in dimension $$W_k({\mathbf A})=\left\{(\alpha_1,\dots,\alpha_m)\in {\mathbb C}^m:\ \alpha_i=\sum_{j=1}^k \langle A_i x_j,x_j\rangle \text{ for an orthonormal set } \{x_1,\dots,x_k\}\subset {\mathcal H}\right\}.$$4 (Chan et al., 2021).

Certain block forms force convexity. If a unitary $$W_k({\mathbf A})=\left\{(\alpha_1,\dots,\alpha_m)\in {\mathbb C}^m:\ \alpha_i=\sum_{j=1}^k \langle A_i x_j,x_j\rangle \text{ for an orthonormal set } \{x_1,\dots,x_k\}\subset {\mathcal H}\right\}.$$5 brings each $$W_k({\mathbf A})=\left\{(\alpha_1,\dots,\alpha_m)\in {\mathbb C}^m:\ \alpha_i=\sum_{j=1}^k \langle A_i x_j,x_j\rangle \text{ for an orthonormal set } \{x_1,\dots,x_k\}\subset {\mathcal H}\right\}.$$6 into one of the block forms

$$W_k({\mathbf A})=\left\{(\alpha_1,\dots,\alpha_m)\in {\mathbb C}^m:\ \alpha_i=\sum_{j=1}^k \langle A_i x_j,x_j\rangle \text{ for an orthonormal set } \{x_1,\dots,x_k\}\subset {\mathcal H}\right\}.$$7

with $$W_k({\mathbf A})=\left\{(\alpha_1,\dots,\alpha_m)\in {\mathbb C}^m:\ \alpha_i=\sum_{j=1}^k \langle A_i x_j,x_j\rangle \text{ for an orthonormal set } \{x_1,\dots,x_k\}\subset {\mathcal H}\right\}.$$8 in the second case, then $$W_k({\mathbf A})=\left\{(\alpha_1,\dots,\alpha_m)\in {\mathbb C}^m:\ \alpha_i=\sum_{j=1}^k \langle A_i x_j,x_j\rangle \text{ for an orthonormal set } \{x_1,\dots,x_k\}\subset {\mathcal H}\right\}.$$9 is convex. These criteria are structural rather than purely dimensional: they arise from convexity of the admissible projection blocks in the compression model (Chan et al., 2021).

Convexity does not propagate monotonically in $W_k({\mathbf A})$0. Example 3.3 shows a tuple built from Pauli blocks for which $W_k({\mathbf A})$1 is not convex while $W_k({\mathbf A})$2 is convex for broad ranges of $W_k({\mathbf A})$3. Nonetheless, monotonicity survives at the level of convex hulls: $W_k({\mathbf A})$4 The geometry of $W_k({\mathbf A})$5 is governed by support half-spaces. For a real unit vector $W_k({\mathbf A})$6, define $W_k({\mathbf A})$7. Then

$W_k({\mathbf A})$8

contains $W_k({\mathbf A})$9, its boundary is a support hyperplane, and

$W_k({\mathbf A})$0

Thus the support function in direction $W_k({\mathbf A})$1 is the sum of the top $W_k({\mathbf A})$2 eigenvalues of the self-adjoint linear combination $W_k({\mathbf A})$3 (Chan et al., 2021).

3. Closure, essential numerical range, and infinite-dimensional phenomena

The finite-dimensional and infinite-dimensional theories diverge most clearly at the level of topology. In finite dimension, $W_k({\mathbf A})$4 is closed. In infinite dimension, closedness can fail even for diagonal operators and even when neighboring ranks behave differently; the paper records examples where $W_k({\mathbf A})$5 is closed while $W_k({\mathbf A})$6 is not (Chan et al., 2021).

The relevant asymptotic object is the joint essential numerical range

$W_k({\mathbf A})$7

where $W_k({\mathbf A})$8 is the ideal of compact operators. An equivalent characterization is that $W_k({\mathbf A})$9 if and only if there exists a weakly null sequence of unit vectors, equivalently an orthonormal sequence, $$W_k({\mathbf A})=\{(\operatorname{Tr}(PA_1),\dots,\operatorname{Tr}(PA_m)):\ P=P^*=P^2,\ \operatorname{rank}P=k\}.$$0 such that

$$W_k({\mathbf A})=\{(\operatorname{Tr}(PA_1),\dots,\operatorname{Tr}(PA_m)):\ P=P^*=P^2,\ \operatorname{rank}P=k\}.$$1

The set $$W_k({\mathbf A})=\{(\operatorname{Tr}(PA_1),\dots,\operatorname{Tr}(PA_m)):\ P=P^*=P^2,\ \operatorname{rank}P=k\}.$$2 is always convex and closed (Chan et al., 2021).

Closure of $$W_k({\mathbf A})=\{(\operatorname{Tr}(PA_1),\dots,\operatorname{Tr}(PA_m)):\ P=P^*=P^2,\ \operatorname{rank}P=k\}.$$3 is described by an essential-diagonal augmentation. For self-adjoint $$W_k({\mathbf A})=\{(\operatorname{Tr}(PA_1),\dots,\operatorname{Tr}(PA_m)):\ P=P^*=P^2,\ \operatorname{rank}P=k\}.$$4, one forms diagonal operators $$W_k({\mathbf A})=\{(\operatorname{Tr}(PA_1),\dots,\operatorname{Tr}(PA_m)):\ P=P^*=P^2,\ \operatorname{rank}P=k\}.$$5 on a Hilbert space with basis indexed by $$W_k({\mathbf A})=\{(\operatorname{Tr}(PA_1),\dots,\operatorname{Tr}(PA_m)):\ P=P^*=P^2,\ \operatorname{rank}P=k\}.$$6, then sets

$$W_k({\mathbf A})=\{(\operatorname{Tr}(PA_1),\dots,\operatorname{Tr}(PA_m)):\ P=P^*=P^2,\ \operatorname{rank}P=k\}.$$7

The resulting identities are

$$W_k({\mathbf A})=\{(\operatorname{Tr}(PA_1),\dots,\operatorname{Tr}(PA_m)):\ P=P^*=P^2,\ \operatorname{rank}P=k\}.$$8

and

$$W_k({\mathbf A})=\{(\operatorname{Tr}(PA_1),\dots,\operatorname{Tr}(PA_m)):\ P=P^*=P^2,\ \operatorname{rank}P=k\}.$$9

Moreover, $k$00 is closed if and only if $k$01, equivalently every point in $k$02 has the form

$k$03

for some positive semidefinite contraction $k$04 with rank at most $k$05 and some $k$06 (Chan et al., 2021).

The convex closure satisfies the exact formula

$k$07

where

$k$08

Consequently, $k$09 is closed if and only if

$k$10

for each $k$11 (Chan et al., 2021).

Closedness has a partial inheritance property. If $k$12 and $k$13 is closed, then $k$14 is closed. If $k$15 and $k$16 is closed, then $k$17 is closed. The general question whether $k$18 closed implies $k$19 closed for $k$20 remains open (Chan et al., 2021).

4. Polyhedrality and structural decomposition

Polyhedrality provides the most rigid geometric regime. In this context, a polyhedral set is the convex hull of finitely many points. The central equivalence for operator tuples states that for $k$21 and fixed $k$22, the following are equivalent: $k$23 is polyhedral for all $k$24; $k$25 is polyhedral; and there exists $k$26 and a unitary $k$27 such that

$k$28

with

$k$29

Thus polyhedrality is equivalent to reduction to a finite-dimensional common reducing subspace carrying diagonal compressions that already generate the full joint $k$30-numerical range (Chan et al., 2021).

A parallel result characterizes polyhedrality of closures. The following are equivalent: $k$31 is polyhedral for all $k$32; $k$33 is polyhedral; and there exist $k$34 and isometries $k$35 such that

$k$36

is a sequence of diagonal $k$37-tuples converging to $k$38, with

$k$39

in the Hausdorff metric. This realizes polyhedral closure as a Hausdorff limit of polyhedra produced by finite-dimensional diagonal compressions (Chan et al., 2021).

Conical boundary points expose the same structure from the boundary inward. If

$k$40

is a conical point of $k$41, where $k$42 is an isometry, then the range of $k$43 is a reducing subspace for each $k$44. If $k$45 is a conical point of $k$46 but $k$47, then $k$48 is approximated by compressions coming from a sequence of isometries $k$49 (Chan et al., 2021).

In the matrix case, polyhedrality also detects commutativity near half-rank. Li, Poon, and Wang proved that a family of $k$50 matrices is a family of mutually commuting normal matrices if and only if $k$51 is polyhedral for some $k$52 satisfying $k$53; equivalently, for a generating family it suffices that $k$54 be polyhedral for any two matrices $k$55 in the family. More generally, they characterized when the joint $k$56-numerical range $k$57 is polyhedral (Li et al., 2020).

5. Commuting normal families and explicit spectral descriptions

For commuting normal operators, joint higher-rank numerical ranges admit explicit spectral models. If $k$58 are commuting normal compact operators, then there is a joint diagonalization

$k$59

and one writes

$k$60

In finite dimension $k$61, if $k$62 denotes the set of diagonal $k$63-projections, then

$k$64

Since

$k$65

one obtains

$k$66

where $k$67 is the set of $k$68-sums of the joint eigenvalue vectors $k$69. In particular, in the commuting normal finite-dimensional case, $k$70 is a polytope whose vertices are the $k$71-sums of joint eigenvalue vectors (Chan et al., 2021).

For compact operators, commuting normality is characterized by these spectral sums. If $k$72 is compact, then $k$73 is a commuting family of normal operators if and only if

$k$74

for every $k$75, where $k$76 is the set of sums of $k$77 joint eigenvalues corresponding to $k$78 linearly independent common eigenvectors. Yet commuting normality does not force polyhedrality in infinite dimension: there exist commuting compact self-adjoint operators $k$79 with $k$80 not polyhedral and with smooth extreme points for all $k$81 (Chan et al., 2021).

For compact tuples, closure is particularly simple: $k$82 and

$k$83

Consequently, for compact operators the following are equivalent: $k$84 is polyhedral for every $k$85; $k$86 is polyhedral for every $k$87; and $k$88 is a commuting family of normal operators with $k$89 polyhedral for every $k$90. In the finite-rank case, commuting normality is equivalent to polyhedrality of $k$91 for all $k$92, and also equivalent to polyhedrality of $k$93 for some $k$94, where $k$95 is the dimension of the sum of the ranges of the $k$96 (Chan et al., 2021).

The matrix-theoretic results of Li, Poon, and Wang sharpen this picture in finite dimensions. For Hermitian tuples they describe $k$97 as the intersection of half-spaces determined by eigenvalues of linear combinations $k$98, and show that conical points of $k$99 force common direct-sum decompositions. When the weight matrix ${\mathcal H}$00 has ${\mathcal H}$01 distinct eigenvalues, the existence of a conical point implies that ${\mathcal H}$02 is a commuting family of normal matrices (Li et al., 2020).

6. Related generalizations, applications, and open problems

Joint higher-rank numerical ranges sit alongside several related constructions. One direction replaces scalar sums over orthonormal ${\mathcal H}$03-frames by compressions to scalar matrices or block-diagonal matricial forms. In the notation of higher-rank matricial ranges,

${\mathcal H}$04

and the joint higher-rank numerical range ${\mathcal H}$05 is its scalar slice: ${\mathcal H}$06 The joint rank ${\mathcal H}$07-matricial range ${\mathcal H}$08 further requires

${\mathcal H}$09

This framework is tied directly to hybrid quantum error correction: for a noisy quantum channel with Kraus operators ${\mathcal H}$10, a hybrid ${\mathcal H}$11 code exists if and only if

${\mathcal H}$12

(Kribs et al., 2019).

Non-emptiness and geometry of these matricial ranges are controlled by dimension bounds. If ${\mathcal H}$13 and ${\mathcal H}$14, then

${\mathcal H}$15

Under the stronger bound

${\mathcal H}$16

${\mathcal H}$17 is star-shaped with star center ${\mathcal H}$18 for any ${\mathcal H}$19. These results extend the higher-rank geometry from scalar compressions to block-diagonal compressions motivated by operator-algebraic quantum error correction (Kribs et al., 2019).

A different extension appears in max algebra. There the paper on generalized numerical ranges in max algebra introduces a rank-${\mathcal H}$20 numerical range, a max joint ${\mathcal H}$21-numerical range, and a max joint ${\mathcal H}$22-numerical range for entry-wise nonnegative matrices. In that setting,

${\mathcal H}$23

and the resulting theory replaces linear algebra over ${\mathcal H}$24 by max-times semiring structure, permutation invariance, and interval-type geometry (Aboutalebi et al., 2024). This suggests that the higher-rank viewpoint is robust under substantial changes in ambient algebraic structure.

Several open problems remain explicit in the operator-theoretic theory. Chan, Li, and Poon list the following: prove or disprove

${\mathcal H}$25

prove or disprove that ${\mathcal H}$26 is closed whenever ${\mathcal H}$27 is closed for ${\mathcal H}$28; construct ${\mathcal H}$29 with ${\mathcal H}$30 closed but ${\mathcal H}$31 not closed for ${\mathcal H}$32; and extend the theory to joint ${\mathcal H}$33-numerical ranges ${\mathcal H}$34 (Chan et al., 2021). These questions mark the boundary between currently understood spectral-compression phenomena and the unresolved geometry of multivariable higher-rank ranges.